Theorem upfval 49301

Description: Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) (Proof shortened by Zhi Wang, 12-Nov-2025.)

Hypotheses

Ref	Expression
upfval.b	⊢ 𝐵 = (Base‘𝐷)
upfval.c	⊢ 𝐶 = (Base‘𝐸)
upfval.h	⊢ 𝐻 = (Hom ‘𝐷)
upfval.j	⊢ 𝐽 = (Hom ‘𝐸)
upfval.o	⊢ 𝑂 = (comp‘𝐸)

Assertion

Ref	Expression
upfval	⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Distinct variable groups:   𝐵,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐶,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝐷,𝑓,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐸,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐻,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝐽,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦   𝑓,𝑂,𝑔,𝑘,𝑚,𝑤,𝑥,𝑦

Proof of Theorem upfval

Dummy variables 𝑏 𝑐 𝑑 𝑒 ℎ 𝑗 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6843	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) ∈ V)
2		fveq2 6828	. . . . . 6 ⊢ (𝑑 = 𝐷 → (Base‘𝑑) = (Base‘𝐷))
3	2	adantr 480	. . . . 5 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = (Base‘𝐷))
4		upfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
5	3, 4	eqtr4di 2786	. . . 4 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → (Base‘𝑑) = 𝐵)
6		fvexd 6843	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) ∈ V)
7		simplr 768	. . . . . . 7 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑒 = 𝐸)
8	7	fveq2d 6832	. . . . . 6 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = (Base‘𝐸))
9		upfval.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝐸)
10	8, 9	eqtr4di 2786	. . . . 5 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (Base‘𝑒) = 𝐶)
11		fvexd 6843	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) ∈ V)
12		simplll 774	. . . . . . . 8 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → 𝑑 = 𝐷)
13	12	fveq2d 6832	. . . . . . 7 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = (Hom ‘𝐷))
14		upfval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐷)
15	13, 14	eqtr4di 2786	. . . . . 6 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → (Hom ‘𝑑) = 𝐻)
16		fvexd 6843	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) ∈ V)
17		simp-4r 783	. . . . . . . . 9 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → 𝑒 = 𝐸)
18	17	fveq2d 6832	. . . . . . . 8 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = (Hom ‘𝐸))
19		upfval.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐸)
20	18, 19	eqtr4di 2786	. . . . . . 7 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → (Hom ‘𝑒) = 𝐽)
21		fvexd 6843	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) ∈ V)
22		simp-5r 785	. . . . . . . . . 10 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → 𝑒 = 𝐸)
23	22	fveq2d 6832	. . . . . . . . 9 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = (comp‘𝐸))
24		upfval.o	. . . . . . . . 9 ⊢ 𝑂 = (comp‘𝐸)
25	23, 24	eqtr4di 2786	. . . . . . . 8 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → (comp‘𝑒) = 𝑂)
26		simp-6l 786	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑑 = 𝐷)
27		simp-6r 787	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑒 = 𝐸)
28	26, 27	oveq12d 7370	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑑 Func 𝑒) = (𝐷 Func 𝐸))
29		simp-4r 783	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑐 = 𝐶)
30		simp-5r 785	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑏 = 𝐵)
31	30	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ 𝐵))
32		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑗 = 𝐽)
33	32	oveqd 7369	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑥)) = (𝑤𝐽((1st ‘𝑓)‘𝑥)))
34	33	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥)) ↔ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))))
35	31, 34	anbi12d 632	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥)))))
36	32	oveqd 7369	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑤𝑗((1st ‘𝑓)‘𝑦)) = (𝑤𝐽((1st ‘𝑓)‘𝑦)))
37		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ℎ = 𝐻)
38	37	oveqdr 7380	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
39		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → 𝑜 = 𝑂)
40	39	oveqd 7369	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦)) = (⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦)))
41	40	oveqd 7369	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))
42	41	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
43	38, 42	reueqbidv 3385	. . . . . . . . . . . . 13 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
44	36, 43	raleqbidv 3313	. . . . . . . . . . . 12 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
45	30, 44	raleqbidv 3313	. . . . . . . . . . 11 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚)))
46	35, 45	anbi12d 632	. . . . . . . . . 10 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))))
47	46	opabbidv 5159	. . . . . . . . 9 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})
48	28, 29, 47	mpoeq123dv 7427	. . . . . . . 8 ⊢ (((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) ∧ 𝑜 = 𝑂) → (𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
49	21, 25, 48	csbied2 3883	. . . . . . 7 ⊢ ((((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) ∧ 𝑗 = 𝐽) → ⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
50	16, 20, 49	csbied2 3883	. . . . . 6 ⊢ (((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) ∧ ℎ = 𝐻) → ⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
51	11, 15, 50	csbied2 3883	. . . . 5 ⊢ ((((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) ∧ 𝑐 = 𝐶) → ⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
52	6, 10, 51	csbied2 3883	. . . 4 ⊢ (((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
53	1, 5, 52	csbied2 3883	. . 3 ⊢ ((𝑑 = 𝐷 ∧ 𝑒 = 𝐸) → ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
54		df-up 49299	. . 3 ⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))}))
55		ovex 7385	. . . 4 ⊢ (𝐷 Func 𝐸) ∈ V
56	9	fvexi 6842	. . . 4 ⊢ 𝐶 ∈ V
57	55, 56	mpoex 8017	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) ∈ V
58	53, 54, 57	ovmpoa 7507	. 2 ⊢ ((𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
59		reldmup 49300	. . . 4 ⊢ Rel dom UP
60	59	ovprc 7390	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = ∅)
61		reldmfunc 49200	. . . . . 6 ⊢ Rel dom Func
62	61	ovprc 7390	. . . . 5 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 Func 𝐸) = ∅)
63	62	orcd 873	. . . 4 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → ((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅))
64		0mpo0 7435	. . . 4 ⊢ (((𝐷 Func 𝐸) = ∅ ∨ 𝐶 = ∅) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
65	63, 64	syl 17	. . 3 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) = ∅)
66	60, 65	eqtr4d 2771	. 2 ⊢ (¬ (𝐷 ∈ V ∧ 𝐸 ∈ V) → (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}))
67	58, 66	pm2.61i 182	1 ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))})

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345  Vcvv 3437  ⦋csb 3846  ∅c0 4282  ⟨cop 4581  {copab 5155  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  1^st c1st 7925  2^nd c2nd 7926  Basecbs 17122  Hom chom 17174  compcco 17175   Func cfunc 17763   UP cup 49298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-func 17767  df-up 49299

This theorem is referenced by:  upfval2  49302  uppropd  49306  reldmup2  49307  relup  49308  uprcl  49309